Section 1.4 - Calculus in Polar Coordinates

Section 1.4 - Calculus in Polar Coordinates

Area of a Region in Polar Coordinates

In polar coordinates, the area of a region bounded by a curve given by a polar equation can be found using calculus. This section develops the formula for the area of a region whose boundary is described by a polar function.

• Polar Coordinates: A point in the plane is represented by (r, θ), where r is the distance from the origin and θ is the angle from the positive x-axis.

• Area of a Sector of a Circle: The area A of a sector with radius r and central angle Δθ (in radians) is given by: $A = \frac{1}{2} r^2 \Delta \theta$

• Area Bounded by a Polar Curve: For a curve r = f(θ) from θ = a to θ = b, the area A is: $A = \int_{a}^{b} \frac{1}{2} [f(\theta)]^2 d\theta$

Example 1:

Find the area enclosed by one loop of the four-leaved rose r = \cos 2θ.

• Set up the integral for one loop (from θ = 0 to θ = \frac{\pi}{2}): $A = \int_{0}^{\frac{\pi}{2}} \frac{1}{2} (\cos 2\theta)^2 d\theta$

Example 2:

Find the area of the region that lies inside the circle r = 3\sinθ and outside the cardioid r = 1 + \sinθ.

• Find the points of intersection by solving 3\sinθ = 1 + \sinθ.

• Set up the area as the difference of two integrals: $A = \int_{\alpha}^{\beta} \frac{1}{2} \left( [3\sin\theta]^2 - [1 + \sin\theta]^2 \right) d\theta$

Area Between Two Polar Curves

To find the area between two polar curves r = f(θ) and r = g(θ) (with f(θ) \geq g(θ)), integrate the difference of their squared radii:

• $A = \int_{a}^{b} \frac{1}{2} \left( [f(\theta)]^2 - [g(\theta)]^2 \right) d\theta$

Example 3:

(a) Find all points of intersection of the curves r = \cos 2θ and r = \frac{1}{2}. (b) Write an integral expression for the area of the region inside both curves.

• Set \cos 2θ = \frac{1}{2} and solve for θ to find intersection points.

• Set up the area integral using the appropriate limits and the formula above.

Arc Length in Polar Coordinates

The length of a curve given by r = f(θ) from θ = a to θ = b is found using the following formula:

• $L = \int_{a}^{b} \sqrt{ [f(\theta)]^2 + \left( \frac{dr}{d\theta} \right)^2 } d\theta$

Example 4:

Find the arc length of the cardioid $r=2+\frac{2}{\cos\theta}$for$-\pi\le\theta\le\pi$.

• Compute$\frac{dr}{d\theta}$and substitute into the arc length formula above.

Summary Table: Key Polar Area and Arc Length Formulas

Concept	Formula	Description
Area inside one curve	$A = \int_{a}^{b} \frac{1}{2} [f(\theta)]^2 d\theta$	Area bounded by r = f(θ) from θ = a to θ = b
Area between two curves	$A = \int_{a}^{b} \frac{1}{2} \left( [f(\theta)]^2 - [g(\theta)]^2 \right) d\theta$	Area between r = f(θ) and r = g(θ)
Arc length	$L = \int_{a}^{b} \sqrt{ [f(\theta)]^2 + \left( \frac{dr}{d\theta} \right)^2 } d\theta$	Length of curve r = f(θ) from θ = a to θ = b

Additional info: The examples provided illustrate the application of these formulas to specific polar curves, such as roses, circles, and cardioids. When setting up integrals, always determine the correct limits of integration by finding intersection points or the range for one complete loop.